Extended Weak Convergence and Utility Maximization with Proportional Transaction Costs
aa r X i v : . [ q -f i n . M F ] J u l EXTENDED WEAK CONVERGENCE AND UTILITYMAXIMIZATION WITH PROPORTIONAL TRANSACTIONCOSTS
ERHAN BAYRAKTAR , LEONID DOLINSKYI, AND YAN DOLINSKY
Abstract.
In this paper we study utility maximization with proportionaltransaction costs. Assuming extended weak convergence of the underlyingprocesses we prove the convergence of the corresponding utility maximizationproblems. Moreover, we establish a limit theorem for the optimal tradingstrategies. The proofs are based on the extended weak convergence theorydeveloped in [1] and the Meyer–Zheng topology introduced in [24].
Contents
1. Introduction 12. Preliminaries and Assumptions 22.1. Hedging with Proportional Transaction Costs 22.2. Approximating Sequence of Models 42.3. Extended Weak Convergence 63. Main Results 64. Proof of the Main Results 84.1. Three Crucial Lemmata 84.2. Completion of the Proof of Theorems 3.1–3.2. 135. Example: Transaction Costs Make Things Converge 16References 171.
Introduction
We deal with the continuity of the utility maximization problem in the presenceof proportional transaction costs, under convergence in distribution of the financialmarkets. More specifically, we focus on utility maximization from terminal wealthunder an admissibility condition on the wealth processes. Although the problem ofutility maximization with proportional transaction costs was widely studied (see,for instance, [22, 28, 15, 13, 27, 8, 16, 17, 3, 21, 9, 18, 11, 7]), to the best of ourknowledge, the continuity under weak convergence was not considered before.Clearly, the problem of utility maximization depends on the flow of information(filtration). Hence, one should not expect that convergence of asset prices alone willimply the convergence of the corresponding control problems. In particular, there
Mathematics Subject Classification.
Key words and phrases.
Utility Maximization, Proportional Transaction Costs, ExtendedWeak Convergence, Meyer–Zheng Topology .Y. Dolinsky is supported by the GIF Grant 1489-304.6/2019 and the ISF grant 160/17. are many examples (see for instance page 75 in [1]) of processes which are “close”to each other in distribution but the behaviour of the corresponding filtrations iscompletely different. This brings us to a stronger notion of convergence.In his important unpublished manuscript [1], Aldous introduced the notion ofextended weak convergence and showed that for optimal stopping this is the rightnotion of convergence. Extended weak convergence is defined via weak convergenceof the corresponding prediction processes. The prediction process is a measurevalued process representing the regular conditional distributions of the originalstochastic process, and so it captures the structure of the information flow. It isimportant to mention the recent papers [4, 5] which provide a novel approach inthe discrete time setup to weak convergence topologies that take the informationflow into account.In this work we consider a sequence of continuous time financial markets withcontinuous asset prices which converge to a limit market. To ease notations we focuson the case of one risky asset. Our main assumptions are extended weak convergenceof the underlying processes and an existence of strictly consistent price systems.Under these natural conditions we prove that for a continuous and concave utilityfunction, there is a convergence of expected utilities; see Theorem 3.1. Moreover,we obtain a limit theorem for the optimal trading strategies; see Theorem 3.2. Thispaper is the first work that applies extended weak convergence to continuous timeportfolio optimization problems.In addition to the extended weak convergence, we also apply the Meyer–Zhengtopology which was introduced in [24]. Roughly speaking, the Meyer–Zheng topol-ogy on the set of functions is the topology of convergence in measure. As we will see,this topology perfectly fits for hedging with proportional transaction costs. Moreprecisely, the admissibility condition will imply tightness of the trading strategiesin the Meyer–Zheng topology.The current work is a continuation of [6] where a similar problem was studiedin the frictionless setup (i.e. with no transaction costs). Surprisingly, for thefrictionless setup extended weak convergence is not a sufficient assumption, and inorder to have convergence for the expected utilities one needs to require convergenceof the equivalent martingale measures (see Assumption 2.5 in [6]). These objectscan be viewed as consistent price systems for the frictionless case. It is importantto emphasize that with the presence of proportional transaction costs there is noneed to assume any convergence structure on the consistent price systems. Namely,the presence of proportional transaction costs provides the needed compactness.The rest of the paper is organized as follows. In the next section we introducethe setup and list our assumptions. In Section 3, we formulate our main results,which are proved in Section 4. In Section 5, we provide a specific example forfinancial markets which converge to a stochastic volatility model. We show that inthe presence of proportional transaction costs the expected utilities converge, whilein the frictionless setup there is no convergence.2. Preliminaries and Assumptions
Hedging with Proportional Transaction Costs.
We consider a modelwith one risky asset which we denote by S = ( S t ) ≤ t ≤ T , where T < ∞ is thetime horizon. We assume that the investor has a bank account that, for simplicity,bears no interest. The process S is assumed to be an adapted, strictly positive and continuous process (not necessarily a semi-martingale) defined on a filteredprobability space (Ω , F , ( F t ) ≤ t ≤ T , P ) where the filtration ( F t ) ≤ t ≤ T satisfies theusual assumptions (right continuity and completeness).Let κ ∈ (0 ,
1) be a constant. Consider a model in which every purchase or sale ofthe risky asset at time t ∈ [0 , T ] is subject to a proportional transaction cost of rate κ . A trading strategy is an adapted process γ = ( γ t ) ≤ t ≤ T of bounded variationwith right continuous paths. The random variable γ t denotes the number of sharesat time t . We use the convention γ − = 0. Moreover, we require that γ T = 0 whichmeans that we liquidate the portfolio at the maturity date.Let γ t := γ + t − γ − t , t ∈ [0 , T ] be the Jordan decomposition into a positivevariation ( γ + t ) ≤ t ≤ T and a negative variation ( γ − t ) ≤ t ≤ T . Since the bid price processis (1 − κ ) S and the ask price process is (1+ κ ) S , then the portfolio value of a tradingstrategy γ at time t is given by V γt := (1 − κ ) Z t S u dγ − u − (1 + κ ) Z t S u dγ + u + (1 − κ ) S t ( γ t ) + − (1 + κ ) S t ( γ t ) − where all the above integrals are pathwise Riemann–Stieltjes integrals. We noticethat the integrals take into account the possible transaction at t = 0. By rearrangingthe terms we get:(2.1) V γt = γ t S t − Z t S u dγ u − κ | γ t | S t − κ Z t S u | dγ u | , t ∈ [0 , T ] . Observe that the wealth process ( V γt ) ≤ t ≤ T is RCLL (right continuous with leftlimits) and from the fact that γ T = 0 it follows that V γT − = V γT . For any initialcapital x > A ( x ) the set of all trading strategies γ which satisfythe admissibility condition x + V γt ≥
0, for all t ∈ [0 , T ].Next, we introduce our utility maximization problem. Let C [0 , T ] be the spaceof all continuous functions f : [0 , T ] → R equipped with the uniform topology.Consider a continuous function U : (0 , ∞ ) × C [0 , T ] → R such that for any s ∈ C [0 , T ] the function U ( · , s ) is non–decreasing and concave. We extend U to R + × C [0 , T ] by U (0 , s ) := lim v ↓ U ( v, s ) (the limit might be −∞ ). Assumption 2.1. (i) For any x > we have E P [ U ( x, S )] > −∞ .(ii) There exist continuous functions m , m : [0 , → R + with m (0) = m (0) = 0 (modulus of continuity) and a non-negative random variable ζ ∈ L (Ω , F , P ) suchthat for any λ ∈ (0 , and v > U ((1 − λ ) v, S ) ≥ (1 − m ( λ )) U ( v, S ) − m ( λ ) ζ. For a given initial capital x > u ( x ) := sup γ ∈ A ( x ) E P [ U ( x + V γT , S )] , where, for a random variable X which satisfies E P [max( − X, ∞ we set E P [ X ] := −∞ . Let us notice that Assumption 2.1(i) implies u ( x ) > −∞ . Remark 2.1.
We should note that the power and the log utility satisfy Assump-tion 2.1. Moreover, for a continuous function g : R → R + the utility U ( v, s ) := − ( g ( s T ) − v ) + which corresponds to shortfall risk minimization for a vanilla optionwith the payoff g ( S T ) , also satisfies Assumption 2.1 (provided that E P [ g ( S T )] < ∞ ). Indeed, in this case, if v ≥ g ( S T )1 − λ then U ((1 − λ ) v, S ) = U ( v, S ) = 0 . If v < g ( S T )1 − λ ,then | U ((1 − λ ) v, S ) − U ( v, S ) | ≤ λv ≤ λ − λ g ( S T ) . Thus, for m ( λ ) := 0 , m ( λ ) := λ − λ and ζ := g ( S T ) , Assumption 2.1 holds true. Approximating Sequence of Models.
For any n , let S ( n ) = ( S ( n ) t ) ≤ t ≤ T .be a strictly positive, continuous process defined on some filtered probability space(Ω n , F ( n ) , ( F ( n ) t ) ≤ t ≤ T , P n ), the filtration ( F ( n ) t ) ≤ t ≤ T satisfies the usual assump-tions . For the n –th model, a trading strategy is a right continuous adapted pro-cesses γ ( n ) = ( γ ( n ) t ) ≤ t ≤ T of bounded variation which satisfies γ ( n ) T = 0. As before,we use the convention that γ ( n )0 − = 0. The corresponding portfolio value is given by V γ ( n ) t := γ ( n ) t S ( n ) t − Z t S ( n ) u dγ ( n ) u − κ | γ ( n ) t | S ( n ) t − κ Z t S ( n ) u | dγ ( n ) u | , t ∈ [0 , T ] . For any x > A ( n ) ( x ) the set of all trading strategies γ ( n ) whichsatisfy x + V γ ( n ) t ≥
0, for all t ∈ [0 , T ]. Set u n ( x ) := sup γ ( n ) ∈ A ( n ) ( x ) E P n h U (cid:16) x + V γ ( n ) T , S ( n ) (cid:17)i . The following assumption will be essential for proving tightness and some inte-grability properties of the admissible trading strategies.
Assumption 2.2.
There exists ε ∈ (0 , κ ) and probability measures Q ∼ P , Q n ∼ P n , n ∈ N with the following properties:1)There exists Q local–martingale ( M t ) ≤ t ≤ T and for any n ∈ N there exists a Q n local–martingale ( M ( n ) t ) ≤ t ≤ T such that | M − S | ≤ ( κ − ε ) S, | M ( n ) − S ( n ) | ≤ ( κ − ε ) S ( n ) , ∀ n ∈ N .
2) The sequence of probability measures P n , n ∈ N , is contiguous to the sequence Q n , n ∈ N . Namely, for any sequence of events A n ∈ F ( n ) , n ∈ N if lim n →∞ Q n ( A n ) =0 then lim n →∞ P n ( A n ) = 0 . Remark 2.2.
Assumption 2.2 can be viewed as a uniform version of the existenceof strictly consistent price systems. For a given model the existence of a strictlyconsistent price system is equivalent to robust no free lunch with vanishing riskcondition for simple strategies; see e.g. [19].
Next, we assume the following uniform integrability assumptions.
Assumption 2.3. (i) For any x > the set { U − ( x, S ( n ) ) } n ∈ N is uniformly integrable where U − :=max( − U, .(ii) For any x > the set n U + (cid:16) x + V γ ( n ) T , S ( n ) (cid:17)o n ∈ N ,γ ( n ) ∈ A ( n ) ( x ) is uniformlyintegrable, where U + := max( U, . Notice, that from Assumption 2.3 −∞ < lim inf n →∞ u n ( x ) ≤ lim sup n →∞ u n ( x ) < ∞ , ∀ x > . In general, if U is not bounded from above, then the verification of Assumption2.3(ii) can be a difficult task. The following result gives quite general and easilyverifiable conditions which are sufficient for Assumption 2.3 to hold true. Proposition 2.1.
Suppose there exist constants
L > , < α < and q > − α which satisfy the following.(I) For all ( v, s ) ∈ (0 , ∞ ) × C [0 , T ] , U ( v, s ) ≤ L (1 + v α ) . (II) For any n ∈ N there exists a probability measure ˆ Q n ∼ P n , a ˆ Q n local–martingale ( ˆ M ( n ) t ) ≤ t ≤ T such that | ˆ M ( n ) − S ( n ) | ≤ κS ( n ) (i.e. ( ˆ M ( n ) , ˆ Q n ) is aconsistent price system) and sup n ∈ N E ˆ Q n h(cid:16) d P n d ˆ Q n (cid:17) q i < ∞ . Then Assumption 2.3(ii) holds true.Proof.
The proof is done by using similar arguments as in Lemma 2.2 in [6]. Theonly needed property is that for any n ∈ N and γ ( n ) ∈ A n ( x ) we have E ˆ Q n [ V γ ( n ) T ] ≤
0. This is a well known property of consistent price systems (see, for instance,Proposition 1.6 in [25]). (cid:3)
We end this section with an example of market models which satisfy Assumptions2.2–2.3.
Example 2.1.
Assume that for any n the process ( S ( n ) t ) ≥ t ≤ T is given by the SDE (2.2) dS ( n ) t = S ( n ) t (cid:16) µ ( n ) t dt + σ ( n ) t dW ( n ) t (cid:17) , t ∈ [0 , T ] where W ( n ) is a Brownian motion and the processes µ ( n ) , σ ( n ) are predictable, withrespect to F ( n ) . Similarly, assume that dS t = S t ( µ t dt + σ t dW t ) , t ∈ [0 , T ] where W is a Brownian motion and the processes µ, σ are predictable, with respectto F . Moreover, we assume that σ, σ ( n ) are invertible and there exists a constant C (which does not depend on n ) such that (2.3) sup ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12) µ t σ t (cid:12)(cid:12)(cid:12)(cid:12) , sup ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ ( n ) t σ ( n ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C a.s ∀ n ∈ N . Then, from the Girsanov theorem it follows that Assumption 2.2(1) holds true for M := S , M ( n ) := S ( n ) , n ∈ N and d Q d P := exp (cid:18) − R T µ t σ t dW t − R T (cid:16) µ t σ t (cid:17) dt (cid:19) , d Q n d P n := exp − R T µ ( n ) t σ ( n ) t dW ( n ) t − R T (cid:18) µ ( n ) t σ ( n ) t (cid:19) dt ! , n ∈ N . From (2.3) we have sup n ∈ N E Q n (cid:20)(cid:18) d P n d Q n (cid:19) q (cid:21) < ∞ , ∀ q ∈ R , and so Assumption 2.2(2) holds true. Moreover, in view of Proposition 2.1, itfollows that Assumption 2.3 holds true provided that there exists L, α > such that U ( v, s ) ≤ L (1 + v α ) , ∀ ( v, s ) ∈ (0 , ∞ ) × C [0 , T ] . Another type of example can be obtained by linearly interpolating discrete timeprocesses, which are discrete time analogs of (2.2). In Section 5 we provide adetailed analysis of a particular example of this type.
Extended Weak Convergence.
We start with formulating our convergenceassumptions.
Assumption 2.4.
For any k ∈ N let D ([0 , T ]; R k ) be the space of all RCLL func-tions f : [0 , T ] → R k equipped with the Skorokhod topology (for details see [2]).We assume that there exists m ∈ N and a stochastic processes X ( n ) : Ω n → D ([0 , T ]; R m ) , n ∈ N , X : Ω → C ([0 , T ]; R m ) (i.e. X is continuous) which sat-isfy the following:(i) The filtrations ( F ( n ) t ) ≤ t ≤ T , n ∈ N and ( F t ) ≤ t ≤ T , are the usual filtrationsgenerated by X ( n ) , n ∈ N and X , respectively.(ii) We have the weak convergence ( S ( n ) , X ( n ) ) ⇒ ( S, X ) on D ([0 , T ]; R m +1 ) . (iii) We have the extended weak convergence X ( n ) ⇛ X . This means that for any k and a continuous bounded function ψ : D ([0 , T ]; R m ) → R k we have ( X ( n ) , Y ( n ) ) ⇒ ( X, Y ) on D ([0 , T ]; R m + k ) , where Y ( n ) t := E P n (cid:16) ψ ( X ( n ) ) | F ( n ) t (cid:17) and Y t := E P ( ψ ( X ) | F t ) , t ∈ [0 , T ] . Remark 2.3.
In [1] Aldous introduced the notion of “extended weak convergence”via prediction processes. He proved that extended weak convergence is equivalent toa more elementary condition which does not require the use of prediction processes(see [1], Proposition 16.15). This is the definition that we use above.The verification of extended weak convergence was studied in [1] and [20]. CitingAldous (page 130 in [1]) “any weak convergence result proved by the martingale tech-nique can be improved to extended weak convergence”. In particular if the processes X ( n ) , n ∈ N have independent increments and X is continuous in probability thenthe weak convergence X ( n ) ⇒ X implies the extended weak convergence X ( n ) ⇛ X (see Proposition 20.18 in [1] and Corollary 2 in [20]). For more results related toextended weak convergence see [20]. Main Results
We are ready to state our first limit theorem.
Theorem 3.1.
Under Assumptions 2.1-2.4 we have that u ( x ) = lim n →∞ u n ( x ) , for any x > . A natural question is whether we have some kind of convergence for the optimaltrading strategies (optimal controls) as well. In order to formulate our limit theoremfor the optimal controls we need some preparations.Any function f ∈ D [0 , T ] := D ([0 , T ]; R ) can be extended to a function f : R + → R by f ( t ) := f ( T ) for all t ≥ T . The Meyer–Zheng topology, introduced in [24], is arelative topology, on the image measures on graphs ( t, f ( t )) of trajectories t → f ( t ), t ∈ R + under the measure λ ( dt ) := e − t dt (called pseudo-paths), induced by the weak topology on probability laws on the compactified space [0 , ∞ ] × R . FromLemma 1 in [24] it follows that the Meyer–Zheng topology on the space D [0 , T ] isgiven by the metric d MZ ( f, g ) := Z T min(1 , | f ( t ) − g ( t ) | ) dt + | f ( T ) − g ( T ) | , f, g ∈ D [0 , T ] . We denote the corresponding space by D MZ [0 , T ]. Remark 3.1.
In Lemma 1 in [24] the authors proved that the Meyer–Zheng topol-ogy on the space D [0 , ∞ ) is equivalent to convergence in measure. Since in oursetup the functions are constants on the time interval [ T, ∞ ) then convergence inmeasure is given by the above d MZ metric. Now we are ready to formulate our second limit theorem.
Theorem 3.2.
Assume that Assumptions 2.1-2.4 hold. Let x > and ˆ γ ( n ) ∈ A ( n ) ( x ) , n ∈ N be a sequence of asymptotically optimal portfolios, namely (3.4) lim n →∞ (cid:16) u n ( x ) − E P n h U (cid:16) x + V ˆ γ ( n ) T , S ( n ) (cid:17)i(cid:17) = 0 . Then the sequence (of laws) ( S ( n ) , X ( n ) , ˆ γ ( n ) ) , n ∈ N is tight on the space D ([0 , T ]; R m ) × D MZ [0 , T ] , and thanks to Prohorov’s theorem (see [2]), it is relatively compact.Moreover, any cluster point of the sequence ( S ( n ) , X ( n ) , ˆ γ ( n ) ) , n ∈ N (there is atleast one) is of the form ( S, X, ˆ γ ) . Define (3.5) ˆ γ F t := E P (ˆ γ t | F t ) , t ∈ [0 , T ] where, as before ( F t ) ≤ t ≤ T is the usual filtration generated by X and (with abuse ofnotations) E P denotes the expectation on the corresponding probability space. Then, ˆ γ F = (ˆ γ F t ) ≤ t ≤ T is an optimal portfolio, i.e. ˆ γ F ∈ A ( x ) and u ( x ) = E P h U (cid:16) x + V ˆ γ F T , S (cid:17)i . We end this section with the following remark about Theorem 3.2.
Remark 3.2.
In view of Assumption 2.4(ii) any cluster point of the sequence ( S ( n ) , X ( n ) , ˆ γ ( n ) ) , n ∈ N has to be of the form ( S, X, ˆ γ ) . We can show that for asuch cluster point ( S, X, ˆ γ ) , ˆ γ is an optimal portfolio. However, ˆ γ is not necessarilyadapted to the filtration generated by X .One possible way to treat this issue is to follow the weak formulation setup.Roughly speaking, the weak formulation allows the investor to randomize from thestart, and so the filtration is rich enough in the sense that the law of any clusterpoint ( S, X, ˆ γ ) can be represented with an adapted process ˆ γ . On the other hand,since the enlarged filtration does not provide any additional (in comparison with theoriginal filtration) information about the future, the value of the (concave) utilitymaximization problem remains the same as in the original setup. For more detailssee [10].In this paper we do not consider the weak formulation approach, instead we solvethe measurability issue by projecting the limit portfolio ˆ γ on the investor’s filtration ( F t ) ≤ t ≤ T . We will prove that the projection is well defined and gives an optimaltrading strategy.Let us remark that if we had uniqueness results for the optimal trading strat-egy then we could prove that the sequence ( S ( n ) , X ( n ) , ˆ γ ( n ) ) , n ∈ N converges to ( S, X, ˆ γ ) where ˆ γ is the unique optimal control and in particular it is adapted tothe filtration generated by X . Surprisingly, up to date, there are no results relatedto the uniqueness of the optimal trading strategy. Of course, for strictly concaveutility we can prove the uniqueness of the optimal terminal wealth but this does notimply the uniqueness of the optimal trading strategy (see Remark 6.9 in [26]). Thelatter is an interesting question which is left for future research. Proof of the Main Results
Three Crucial Lemmata.
We start with the following result. Recall thatin view of Assumption 2.1(i) u ( x ) > −∞ . Lemma 4.1.
The function u : (0 , ∞ ) → R ∪ {∞} is continuous. Namely, for any x > we have u ( x ) = lim y → x u ( y ) where a priori the joint value can be equal to ∞ .Proof. The proof is done by using similar arguments as in Lemma 2.1 in [6] for thefrictionless case. The only needed property is that for any λ > γ we have the equality V λγt = λV γt , t ∈ [0 , T ]. Trivially, this propertyholds true in our setup. (cid:3) Now, we prove the lower bound part in Theorem 3.1.
Lemma 4.2.
For any x > we have u ( x ) ≤ lim inf n →∞ u n ( x ) . Proof. Step 1.
Let x >
0. Without loss of generality (by passing to a subsequence)we assume that lim n →∞ u n ( x ) exists. In view of Lemma 4.1 in order to prove thestatement, it is sufficient to prove that for any ε ∈ (0 , x/
3) and γ ∈ A ( x − ε ) wehave(4.6) E P [ U ( x + V γT − ε, S )] ≤ lim n →∞ u n ( x ) . From the Skorohod representation theorem (Theorem 3 in [12]) and Assumption2.4(ii), we can redefine the stochastic processes ( S ( n ) , X ( n ) ), n ∈ N and ( S, X ) onthe same probability space such that(4.7) ( S ( n ) , X ( n ) ) → ( S, X ) a.s. on D ([0 , T ]; R m +1 ) . Choose ε ∈ (0 , x/
3) and γ ∈ A ( x − ε ). We aim to prove (4.6). Step 2.
For any n ∈ N , let Γ ( n ) be the set of all trading strategies (do not haveto satisfy admissibility conditions) in the n –step model. First, we show that thereexists a subsequence γ ( n ) ∈ Γ ( n ) , n ∈ N (for simplicity the subsequence is stilldenoted by n ) such that(4.8) V γT ≤ lim inf n →∞ V γ ( n ) T and(4.9) lim inf n →∞ inf ≤ t ≤ T V γ ( n ) t ≥ ε − x. To that end, for any n ∈ N define the predictable (with respect to ( F t ) ≤ t ≤ T )process ¯ γ ( n ) = (¯ γ ( n ) t ) ≤ t ≤ T by¯ γ ( n ) t := n − X i =1 γ ( i − Tn I iTn ≤ t< ( i +1) Tn , t ∈ [0 , T ] . We made small shift in time in order to make ¯ γ ( n ) predictable. Clearly, V γT = − Z T S t dγ t − κ Z T S t | dγ t |≤ lim inf n →∞ − Z T S t d ¯ γ ( n ) t − κ Z T S t | d ¯ γ ( n ) t | ! = lim inf n →∞ V ¯ γ ( n ) T . (4.10)Fix n ∈ N , k = 0 , , ..., n − t ∈ [ kTn , ( k +1) Tn ). From (2.1) and the simplerelations Z iTn ( i − T/n dγ u = Z ( i +1) TniTn d ¯ γ ( n ) u , Z iTn ( i − T/n | dγ u | ≥ Z ( i +1) TniTn | d ¯ γ ( n ) u | , i = 1 , ..., k, it follows that V γ kTn − V ¯ γ ( n )( k +1) Tn ≤ | γ kTn || S ( k +1) Tn − S kTn | + 2 Z kTn | dγ u | sup ≤ t ,t ≤ ( k +1) Tn , | t − t |≤ kTn | S t − S t | . Moreover, since ¯ γ ( n ) is (a random) constant on the interval [ kTn , t ], then from (2.1) V ¯ γ ( n ) kTn − V ¯ γ ( n ) t ≤ | ¯ γ ( n ) kTn || S t − S kTn | . Thus, for any n ∈ N (notice that V ¯ γ ( n ) t = 0 for t < T /n )inf ≤ t ≤ T V ¯ γ ( n ) t ≥ min ≤ k ≤ n V γ kTn − Z T | dγ u | sup | t − t |≤ kTn | S t − S t | . Since γ = A ( x − ε ), then we conclude(4.11) lim inf n →∞ inf ≤ t ≤ T V ¯ γ ( n ) t ≥ inf ≤ t ≤ T V γt ≥ ε − x. Next, let ˜Γ be the set of all simple integrands of the from(4.12) ˜ γ t = k X i =1 β i I t i ≤ t Next, we modify the trading strategies γ ( n ) ∈ Γ n , n ∈ N in order to meetthe admissibility requirements. For any n ∈ N define the stopping time τ n := T ∧ inf { t : x + V γ ( n ) t < ε } , and consider the trading strategy β ( n ) t := γ ( n ) t I t<τ n , t ∈ [0 , T ] . From (2.1) and the definition of τ n it follows that for any n ∈ N (4.17) V β ( n ) t = V γ ( n ) t I t<τ n + V γ ( n ) τ n − I t ≥ τ n ≥ ε − x, t ∈ [0 , T ] . Thus, β ( n ) ∈ A ( n ) ( x ), n ∈ N .From (4.9) we have(4.18) I τ n = T → T . Thus, from (4.8) and(4.17)–(4.18)(4.19) lim inf n →∞ V β ( n ) T ≥ V γT . Finally, from Fatou’s Lemma, Assumption 2.3(i), the inequality x + V β ( n ) T ≥ ε ,(4.7) and (4.19) we obtainlim n →∞ u n ( x ) ≥ lim inf n →∞ E P n h U (cid:16) x + V β ( n ) T , S ( n ) (cid:17)i ≥ E P [ U ( x + V γT − ε, S )]and (4.6) follows. (cid:3) Next, we prove the following key result. Lemma 4.3. Let x > and γ ( n ) ∈ A ( n ) ( x ) , n ∈ N be a sequence of admissible trad-ing strategies. The sequence ( X ( n ) , S ( n ) , γ ( n ) ) is tight on the space D ([0 , T ]; R m +1 ) × D MZ [0 , T ] and so from Prohorov’s theorem (see [2]) it is relatively compact. More-over, any cluster point is of the form ( X, S, γ ) and satisfies the following conditionalindependence property:Let ( F X,γt ) ≤ t ≤ T be the usual filtration generated by X and γ . Then for any t < T , F X,γt and F T are conditionally independent given F t . As before ( F t ) ≤ t ≤ T is the usual filtration generated by X .Proof. Step 1. In [24] (see Lemma 8 there) the authors proved that for any c > ( f : f is of bounded variation and Z T | df ( t ) | ≤ c ) ⊂ D MZ [0 , T ]is compact. Thus, in order to prove tightness (in the Meyer–Zheng topology) of thesequence γ ( n ) ∈ A ( n ) ( x ), n ∈ N , it is sufficient to prove that for any δ > c > N ∈ N such that(4.20) P n Z T | dγ ( n ) t | > c ! < δ, ∀ n > N. Choose δ > 0. Since S is strictly positive then the weak convergence S ( n ) ⇒ S implies that there exists ˆ δ > N ∈ N such that(4.21) P n (cid:18) inf ≤ t ≤ T S ( n ) t < ˆ δ (cid:19) < δ, ∀ n > N. Next, recall the processes M ( n ) , n ∈ N and the probability measures Q n , n ∈ N given by Assumption 2.2. From the inequalities | M ( n ) t − S ( n ) t | , | M ( n ) t − − S ( n ) t | ≤ ( κ − ε ) S ( n ) t , t ∈ [0 , T ] , the admissibility property of γ ( n ) ∈ A ( n ) ( x ), n ∈ N and the integration by partsformula we get0 ≤ x + γ ( n ) t S ( n ) t − Z t S ( n ) u dγ ( n ) u − κ | γ ( n ) t | S ( n ) t − κ Z t S ( n ) u | dγ ( n ) u |≤ x + γ ( n ) t M ( n ) t + ( κ − ε ) | γ ( n ) t | S ( n ) t − Z t M ( n ) u − dγ ( n ) u + ( κ − ε ) Z t S ( n ) u | dγ ( n ) u |− κ | γ ( n ) t | S ( n ) t − κ Z t S ( n ) u | dγ ( n ) u |≤ x + Z t γ ( n ) u − dM ( n ) u − ε Z t S ( n ) u | dγ ( n ) u | , ∀ t ∈ [0 , T ] . Thus, for any n ∈ N (4.22) E Q n "Z T S ( n ) u | dγ ( n ) u | ≤ xε . From the Markov inequality and the fact that the sequence P n , n ∈ N is contiguousto the sequence Q n , n ∈ N we conclude that there exists ˆ c > P n Z T S ( n ) u | dγ ( n ) u | > ˆ c ! < δ, ∀ n ∈ N . This together with (4.21) gives that (4.20) holds true for c := ˆ c ˆ δ and tightnessfollows. Step 2. From Assumption 2.4(ii) we conclude that ( S ( n ) , X ( n ) , γ ( n ) ), n ∈ N istight on the space D ([0 , T ]; R m +1 ) × D MZ [0 , T ] and so from Prohorov’s theorem itis relatively compact. Moreover, from Assumption 2.4(ii) it follows that any clusterpoint is of the form ( S, X, γ ), namely the distribution of the first two componentsequals to the distribution of the pair ( S, X ). The process γ is of bounded variationwith right continuous paths.It remains to establish the conditional independence property. With abuse ofnotations we denote by E P the expectation on the corresponding probability space.Choose t < T . We need to show (see Definition 43 in [14]) that for any boundedrandom variable Z which is F T measurable and a bounded random variable Z which is F X,γt measurable we have the equality E P ( Z Z | F t ) = E P ( Z | F t ) E P ( Z | F t ) . This is equivalent to proving that for any bounded random variable Z which is F t measurable we have E P ( Z Z Z ) = E P ( E P ( Z | F t ) Z Z ) . Since the filtration ( F u ) ≤ u ≤ T is right continuous then the above equality followsfrom the equality(4.23) E P ( Z Z Z ) = E P ( E P ( Z | F u ) Z Z ) , ∀ u > t. Standard density arguments yield that without loss of generality we can assumethat Z = ψ ( X ) for a continuous, bounded function ψ : D ([0 , T ]; R m ) → R . Let Y ( n ) u := E P n (cid:16) ψ ( X ( n ) ) | F ( n ) u (cid:17) and Y u := E P ( ψ ( X ) | F u ) , u ∈ [0 , T ] . By passing to a subsequence, we can assume without loss of generality that (cid:0) S ( n ) , X ( n ) , γ ( n ) (cid:1) converge to ( S, X, γ ). From Assumption 2.4(ii)–(iii) we obtain that the sequence (cid:0) S ( n ) , X ( n ) , Y ( n ) , γ ( n ) (cid:1) , n ∈ N is tight on the space D ([0 , T ]; R m +1 ) × D ([0 , T ]; R ) × D MZ [0 , T ] and so from Prohorov’s theorem it is relatively compact. Moreover,Assumptions 2.4(ii)–(iii) imply that any cluster point is of the form ( S, X, Y, γ ).From the Skorohod representation theorem (Theorem 3 in [12]) there exists aprobability space which contains a subsequence ( S ( n ) , X ( n ) , Y ( n ) , γ ( n ) ), n ∈ N and( S, X, Y, γ ) on the same probability space such that(4.24) ( S ( n ) , X ( n ) , Y ( n ) , γ ( n ) ) → ( S, X, Y, γ ) a.s.on the space D ([0 , T ]; R m +1 ) × D ([0 , T ]; R ) × D MZ [0 , T ].Next, from the bounded convergence theorem we havelim n →∞ E Z T min(1 , | γ ( n ) u − γ u | ) du ! = 0 , where E denotes the expectation on the common probability space. From Fubini’stheorem we conclude that there exists a subset I ⊂ [0 , T ] of a full Lebesgue measuresuch that(4.25) γ u = lim n →∞ γ ( n ) u , ∀ u ∈ I, where the limit is in probability.Choose u > t in (4.23). Again, standard density arguments imply that with-out loss of generality we can assume that Z , Z in (4.23) are of the form: Z = ψ ( X t , ..., X t k ) for some k ∈ N , t , ..., t k ∈ I ∩ [0 , u ] and a continuous boundedfunction ψ : ( R m ) k → R and Z = ψ ( X t , ..., X t k , γ t , ..., γ t k ) for a continuousbounded function ψ : ( R m ) k × R k → R .From the bounded convergence theorem, the fact that γ ( n ) is F ( n ) adapted and(4.24)–(4.25) we obtain E P ( Z Z Z )= lim n →∞ E P n (cid:16) ψ ( X n ) ψ ( X ( n ) t , ..., X ( n ) t k ) ψ ( X ( n ) t , ..., X ( n ) t k , γ ( n ) t , ..., γ ( n ) t k ) (cid:17) = lim n →∞ E P n (cid:16) Y ( n ) u ψ ( X ( n ) t , ..., X ( n ) t k ) ψ ( X ( n ) t , ..., X ( n ) t k , γ ( n ) t , ..., γ ( n ) t k ) (cid:17) = E P ( Y u Z Z ) . This completes the proof of (4.23). (cid:3) Now, we are ready to complete the proof of the main results.4.2. Completion of the Proof of Theorems 3.1–3.2. Proof. We will prove these two results together. Step 1. Let x > γ ( n ) ∈ A ( n ) ( x ), n ∈ N be a sequence of portfolios whichsatisfy (3.4). By passing to a subsequence we assume without loss of generality thatlim n →∞ u n ( x ) exists. From (3.4) lim n →∞ E P n h U (cid:16) x + V ˆ γ ( n ) T , S ( n ) (cid:17)i exists as well.From Lemma 4.3 we obtain that ( S ( n ) , X ( n ) , ˆ γ ( n ) ), n ∈ N is tight and any clusterpoint is of the form ( S, X, ˆ γ ). Obviously ˆ γ is a right continuous process of boundedvariation. Moreover, since ˆ γ ( n ) T = 0 for all n , ˆ γ T = 0 as well. Thus, we define( V ˆ γt ) ≤ t ≤ T by (2.1).Let us prove the admissibility condition(4.26) x + V ˆ γt ≥ , ∀ t ∈ [0 , T ] , and the inequality(4.27) E P [ U ( x + V ˆ γT , S )] ≥ lim n →∞ u n ( x ) . From (4.20) it follows that the sequence S ( n ) , X ( n ) , ˆ γ ( n ) , Z T | d ˆ γ ( n ) t | ! , n ∈ N , is tight on the space D ([0 , T ]; R m +1 ) × D MZ [0 , T ] × R .Thus, by passing to a further subsequence and applying the Skorohod represen-tation theorem we obtain that there exists a common probability space where we have the almost surely convergence(4.28) S ( n ) , X ( n ) , ˆ γ ( n ) , Z T | d ˆ γ ( n ) t | ! → ( S, X, ˆ γ, η ) a.s.for some random variable η < ∞ . We conclude that(4.29) sup n ∈ N Z T | d ˆ γ ( n ) t | < ∞ a.s.Next, using the same arguments as before (4.25) gives that there exists a subset I ⊂ [0 , T ] of a full Lebesgue measure such that ˆ γ ( n ) u → ˆ γ u in probability for all u ∈ I .From a diagonalization argument, it follows that there exists a countable denseset ˆ I and a subsequence ( S ( n ) , X ( n ) , ˆ γ ( n ) ), n ∈ N , such that(4.30) ˆ γ u = lim n →∞ ˆ γ ( n ) u a.s. ∀ u ∈ ˆ I. Since ˆ γ T = 0 and ˆ γ ( n ) T = 0, n ∈ N then without loss of generality we assume that T ∈ ˆ I .Let t ∈ ˆ I . From Theorem 12.16 in [23] and (4.28)–(4.30)(4.31) Z t S u d ˆ γ u = lim n →∞ Z t S ( n ) u d ˆ γ ( n ) u a.s.Next, choose a partition { a < a < ... < a k = t } ⊂ ˆ I . From (4.28)–(4.30)lim inf n →∞ Z t S ( n ) u | d ˆ γ ( n ) u | = lim inf n →∞ Z t S u | d ˆ γ ( n ) u |≥ lim inf n →∞ k − X i =0 min a i ≤ u ≤ a i +1 S u | ˆ γ ( n ) a i +1 − ˆ γ ( n ) a i | = k − X i =0 min a i ≤ u ≤ a i +1 S u | ˆ γ a i +1 − ˆ γ a i | . By taking the mesh of the partition to zero we conclude(4.32) lim inf n →∞ Z t S ( n ) u | d ˆ γ ( n ) u | ≥ Z t S u | d ˆ γ u | a.s.From (4.30)–(4.32)(4.33) V ˆ γt ≥ lim sup n →∞ V ˆ γ ( n ) t ≥ − x a.s. ∀ t ∈ ˆ I. Since ˆ I is a dense set which contains T and ( V ˆ γt ) ≤ t ≤ T is RCLL we obtain (4.26).From Assumption 2.3(ii), (3.4) and (4.33) (for t = T ) we conclude (4.27). Step 2. From the conditional independence property that was proved in Lemma4.3 it follows that any martingale with respect to the filtration ( F t ) ≤ t ≤ T is alsoa martingale with respect to the filtration ( F X, ˆ γt ) ≤ t ≤ T . Thus, we redefine theprobability measure Q and the Q local–martingale M (from Assumption 2.2) onthe common probability space.Since M is a Q local–martingale with respect to the filtration ( F X, ˆ γt ) ≤ t ≤ T , thenby the same arguments as before (4.22), we obtain that (4.26) implies E Q hR T S u | d ˆ γ u | i ≤ xε . This together with the inequality inf ≤ u ≤ T S u > d Q d P is F T measurable gives(4.34) E P Z T | d ˆ γ u | (cid:12)(cid:12)(cid:12)(cid:12) F T ! < ∞ a.s.As usual for any non–negative random variable ξ ≥ σ –algebra G we definethe conditional expectation E ( ξ | G ) as a non–negative extended random variable.For a general random variable ξ we use the convention E ( ξ | G ) := E ( ξ + | G ) − E ( ξ − | G )a.s. on the set where E ( ξ + | G ) , E ( ξ − | G ) < ∞ and E ( ξ | G ) := ∞ on the complement.Let ˆ γ t := ˆ γ + t − ˆ γ − t , t ∈ [0 , T ] be the Jordan decomposition into a positive variation(ˆ γ + t ) ≤ t ≤ T and a negative variation (ˆ γ − t ) ≤ t ≤ T .Define the processesˆ β + t := E P (cid:0) ˆ γ + t | F T (cid:1) , ˆ β − t := E P (cid:0) ˆ γ − t | F T (cid:1) , t ∈ [0 , T ] . From (4.34) we obtain that ( ˆ β + t ) ≤ t ≤ T and ( ˆ β − t ) ≤ t ≤ T are non–decreasing and non–negative processes which satisfy ˆ β + T , ˆ β − T < ∞ a.s. Moreover, from the dominatedconvergence theorem (for conditional expectation), the fact that ˆ γ is right contin-uous and (4.34) it follows that ˆ β + , ˆ β − are right continuous as well.Next, fix t ∈ [0 , T ] and n ∈ N . From [14] (see Chapter 2, Theorem 45) and theconditional independence property that was proved in Lemma 4.3 E P (cid:0) ˆ γ + t ∧ n | F t (cid:1) = E P (cid:0) ˆ γ + t ∧ n | F T (cid:1) . By taking n → ∞ we get(4.35) E P (cid:0) ˆ γ + t | F t (cid:1) = ˆ β + t , ∀ t ∈ [0 , T ] . Similarly,(4.36) E P (cid:0) ˆ γ − t | F t (cid:1) = ˆ β − t , ∀ t ∈ [0 , T ] . Thus, from (3.5)(4.37) ˆ γ F t = ˆ β + t − ˆ β − t = E P (ˆ γ t | F T ) , ∀ t ∈ [0 , T ] . We conclude that (ˆ γ F t ) ≤ t ≤ T is a right continuous process of bounded variation.Clearly, (3.5) implies that (ˆ γ F t ) ≤ t ≤ T is adapted to the filtration ( F t ) ≤ t ≤ T .From (4.34) we have E P (cid:16) sup ≤ t ≤ T S t R T | d ˆ γ u | (cid:12)(cid:12) F T (cid:17) < ∞ a.s. Hence, from thedominated convergence theorem and (4.35)–(4.37) Z t S u d ˆ γ F u = E P (cid:18)Z t S u d ˆ γ u (cid:12)(cid:12) F T (cid:19) , ∀ t ∈ [0 , T ]and Z t S u | d ˆ γ F u | ≤ Z t S u d ˆ β + u + Z t S u d ˆ β − u = E P (cid:18)Z t S u d ˆ γ + u (cid:12)(cid:12) F T (cid:19) + E P (cid:18)Z t S u d ˆ γ − u (cid:12)(cid:12) F T (cid:19) = E P (cid:18)Z t S u | d ˆ γ u | (cid:12)(cid:12) F T (cid:19) , ∀ t ∈ [0 , T ] . This together with (2.1) and the simple relationsˆ γ F t S t = E P (ˆ γ t S t (cid:12)(cid:12) F T ) , | ˆ γ F t S t | ≤ E P ( | ˆ γ t | S t (cid:12)(cid:12) F T ) , ∀ t ∈ [0 , T ] gives(4.38) V ˆ γ F t ≥ E P ( V ˆ γt | F T ) , ∀ t ∈ [0 , T ] , and so from (4.26) ˆ γ F ∈ A ( x ).Thus, from the Jensen inequality, Lemma 4.2, (4.27) and (4.38)lim n →∞ u n ( x ) ≥ u ( x ) ≥ E P [ U ( x + V ˆ γ F T , S )] ≥ E P [ U ( x + V ˆ γT , S )] ≥ lim n →∞ u n ( x ) , and the proof is completed. (cid:3) Example: Transaction Costs Make Things Converge Consider a random utility which corresponds to shortfall risk minimization fora call option with strike price K > 0. Namely, we set U ( v, s ) := − (cid:16) ( s T − K ) + − v (cid:17) + , ∀ ( v, s ) ∈ [0 , ∞ ) × C [0 , T ] . Next, let ξ i = ± i ∈ N be i.i.d. and symmetric. For any n ∈ N define thescaled random walks X n, t , X n, t , t ∈ [0 , T ] by X n, t := r Tn [ nt/T ] X i =1 ξ i ,X n, t := r Tn [ nt/T ] X i =1 i Y j =1 ξ j , where [ · ] is the integer part of · and we set P i =1 ≡ X ( n ) := ( X n, , X n, ) and let ( F ( n ) ) ≤ t ≤ T be the usual filtration generatedby X ( n ) . Observe that X ( n ) is a martingale. From Lemma 3.1 in [6] it follows thatwe have the weak convergence X ( n ) ⇒ X where X = ( X (1) , X (2) ) is a standard twodimensional Brownain motion. From Corollary 6 in [20] we conclude the extendedweak convergence(5.39) X ( n ) ⇛ X. We remark that although [20] deals only with real valued processes, the extensionof the results there to the multidimensional case is straightforward.Now, we introduce the financial markets. For any n ∈ N define the discrete timestochastic processes { ˜ ν ( n ) k } nk =0 and { ˜ S ( n ) k } nk =0 by˜ ν ( n ) k := k Y i =1 r Tn ξ i ! , ˜ S ( n ) k := k Y i =1 (cid:18) ν ( n ) ( i − Tn , ln n (cid:19) r Tn i Y j =1 ξ j , we set Q i =1 ≡ 1. Let S ( n ) = ( S ( n ) t ) ≤ t ≤ t , n ∈ N be the following linear interpolation(5.40) S ( n ) t := (([ nt/T ] + 1) T − nt ) ˜ S ( n )[ nt/T ] − + ( nt − [ nt/T ] T ) ˜ S ( n )[ nt/T ] , where we set ˜ S ( n ) − ≡ 1. We take a shift of one time period in order to make S ( n ) adapted to F ( n ) . For any n define the process ( ν ( n ) t ) ≤ t ≤ T by ν ( n ) t := ˜ ν ( n )[ nt/T ] . Using the same arguments as in Example 3.3 in [6] we obtain the weak conver-gence(5.41) ( ν ( n ) , S ( n ) , X ( n ) ) ⇒ ( ν, S, X ) , where ( ν, S ) is the (unique strong) solution of the SDE (Hull and White model) dν t = ν t dX (1) t , ν = 1 ,dS t = ν t S t dX (2) t , S = 1 . (5.42)Next, we verify Assumptions 2.1-2.4. Clearly, E P S T ≤ S = 1. Thus Assumption2.1(i) holds true. From Remark 2.1 it follows that Assumption 2.1(ii) holds true aswell.Let κ > { ˜ S ( n ) k } nk =0 is amartingale. Thus, for sufficiently large n we obtain that the martingale M ( n ) =( M ( n ) t ) ≤ t ≤ T given by M ( n ) t := ˜ S ( n )[ nt/T ] satisfies | M ( n ) − S ( n ) | ≤ κ S ( n ) . Hence,Assumptions 2.2 holds true for Q n := P n . For the limit model we just take M := S and Q := P .Similar arguments as in Example 3.3 in [6] yield that the random variables { S ( n ) T } n ∈ N are uniformly integrable. This gives Assumption 2.3(i). Since U + = 0then Assumption 2.3(ii) is trivial. Finally, Assumption 2.4 follows from (5.39) and(5.41).From Theorem 3.1 it follows that in the presence of proportional transactioncosts, the shortfall risks in the models given by (5.40) converge to the shortfall riskin the Hull–White stochastic volatility model given by (5.42).On the other hand, for the frictionless case i.e. κ = 0 there is no convergence. Infact, the models given by (5.40) are not arbitrage free. Observe that for any intervalof the form [ kT /n, ( k + 1) T /n ], k ≥ 1, immediately after time kT /n the investorknows all the stock prices in this interval. Thus, we have an obvious arbitrage,which makes the shortfall risk equal to zero for any initial capital. Clearly, this isnot the case for the limit model.Moreover, assume that for the n –step model we restrict the trading times to theset 0 , T /n, T /n, ..., T . We notice that for k = 1 , ..., n − supp (cid:18) S ( n ) ( k +1) Tn | S ( n ) Tn , ..., S ( n ) kTn (cid:19) consists of exactly two points. 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